An Attractive Analytic-Numeric Approach for the Solutions of Uncertain Riccati Differential Equations using Residual Power Series

2020 ◽  
Vol 14 (2) ◽  
pp. 177-190 ◽  
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Riccati Differential Equations ◽  
Residual Power Series ◽  
Residual Power

scholarly journals Computational Optimization of Residual Power Series Algorithm for Certain Classes of Fuzzy Fractional Differential Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Mohammad Alaroud ◽  
Mohammed Al-Smadi ◽  
Rokiah Rozita Ahmad ◽  
Ummul Khair Salma Din
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Optimization Technique ◽  
Fractional Power ◽  
Taylor Formula ◽  
Residual Power Series ◽  
Fractional Differential ◽  
Residual Power ◽  
Fuzzy Fractional Differential Equations

This paper aims to present a novel optimization technique, the residual power series (RPS), for handling certain classes of fuzzy fractional differential equations of order 1<γ≤2 under strongly generalized differentiability. The proposed technique relies on generalized Taylor formula under Caputo sense aiming at extracting a supportive analytical solution in convergent series form. The RPS algorithm is significant and straightforward tool for creating a fractional power series solution without linearization, limitation on the problem’s nature, sort of classification, or perturbation. Some illustrative examples are provided to demonstrate the feasibility of the RPS scheme. The results obtained show that the scheme is simple and reliable and there is good agreement with exact solution.

scholarly journals Residual Power Series Technique for Simulating Fractional Bagley–Torvik Problems Emerging in Applied Physics

2019 ◽  
Vol 9 (23) ◽  
pp. 5029 ◽  
Author(s):  
Alshammari ◽  
Al-Smadi ◽  
Hashim ◽  
Alias
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Nonlinear Modeling ◽  
Test Problems ◽  
Approximate Methods ◽  
Wide Range ◽  
Residual Power Series ◽  
Fractional Differential ◽  
Residual Power

Numerical simulation of physical issues is often performed by nonlinear modeling, which typically involves solving a set of concurrent fractional differential equations through effective approximate methods. In this paper, an analytic-numeric simulation technique, called residual power series (RPS), is proposed in obtaining the numerical solution a class of fractional Bagley–Torvik problems (FBTP) arising in a Newtonian fluid. This approach optimizes the solutions by minimizing the residual error functions that can be directly applied to generate fractional PS with a rapidly convergent rate. The RPS description is presented in detail to approximate the solution of FBTPs by highlighting all the steps necessary to implement the algorithm in addressing some test problems. The results indicate that the RPS algorithm is reliable and suitable in solving a wide range of fractional differential equations applying in physics and engineering.

scholarly journals Applications of General Residual Power Series Method to Differential Equations with Variable Coefficients

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Bochao Chen ◽  
Li Qin ◽  
Fei Xu ◽  
Jian Zu
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Variable Coefficients ◽  
Series Solutions ◽  
Series Method ◽  
Power Series Method ◽  
Residual Power Series Method ◽  
Residual Power Series ◽  
Analytical Series ◽  
Residual Power

This paper is devoted to studying the analytical series solutions for the differential equations with variable coefficients. By a general residual power series method, we construct the approximate analytical series solutions for differential equations with variable coefficients, including nonhomogeneous parabolic equations, fractional heat equations in 2D, and fractional wave equations in 3D. These applications show that residual power series method is a simple, effective, and powerful method for seeking analytical series solutions of differential equations (especially for fractional differential equations) with variable coefficients.

scholarly journals Toward computational algorithm for time-fractional Fokker–Planck models

2019 ◽  
Vol 11 (10) ◽  
pp. 168781401988103 ◽  
Author(s):  
Asad Freihet ◽  
Shatha Hasan ◽  
Mohammad Alaroud ◽  
Mohammed Al-Smadi ◽  
Rokiah Rozita Ahmad ◽  
...  
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Fractional Power ◽  
Series Method ◽  
Power Series Method ◽  
Residual Power Series Method ◽  
Wide Range ◽  
Residual Power Series ◽  
Residual Power ◽  
Fokker Planck

This article describes an efficient algorithm based on residual power series to approximate the solution of a class of partial differential equations of time-fractional Fokker–Planck model. The fractional derivative is assumed in the Caputo sense. The proposed algorithm gives the solution in a form of rapidly convergent fractional power series with easily computable coefficients. It does not require linearization, discretization, or small perturbation. To test simplicity, potentiality, and practical usefulness of the proposed algorithm, illustrative examples are provided. The approximate solutions of time-fractional Fokker–Planck equations are obtained by the residual power series method are compared with those obtained by other existing methods. The present results and graphics reveal the ability of residual power series method to deal with a wide range of partial fractional differential equations emerging in the modeling of physical phenomena of science and engineering.

scholarly journals Numerical Investigation to Fuzzy Volterra Integro-Differential Equations via Residual Power Series Method

2020 ◽  
pp. 1-7
Author(s):  
Mohammad Alshammari ◽  
Mohammed Al-Smadi ◽  
Ishak Hashim ◽  
Mohd Almie Alias
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Effective Technique ◽  
Power Series Method ◽  
Solution Approach ◽  
Residual Power Series Method ◽  
Residual Functions ◽  
Residual Power Series ◽  
Residual Power ◽  
Taylor’S Series

In this paper, a study of a numerical approximate solution to fuzzy Volterra integro-differential equations is presented under strongly generalised differentiability by applying an influent effective technique, called the Residual Power Series (RPS) method. The solution approach can be expressed on Taylor's series formula in terms of elementary σ-level representation, whereas the coefficients can be computed by utilising its residual functions. Furthermore, a numerical computational example is given to test and validate the proposed method. The results reached show several features concerning the RPS method such as potentiality, generality and superiority to handle many problems arising in physics and engineering.

scholarly journals Numerical Investigation to Fuzzy Volterra Integro-Differential Equations via Residual Power Series Method

2020 ◽  
pp. 1-7
Author(s):  
Mohammad Alshammari ◽  
Mohammed Al-Smadi ◽  
Ishak Hashim ◽  
Mohd Almie Alias
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Effective Technique ◽  
Power Series Method ◽  
Solution Approach ◽  
Residual Power Series Method ◽  
Residual Functions ◽  
Residual Power Series ◽  
Residual Power ◽  
Taylor’S Series

In this paper, a study of a numerical approximate solution to fuzzy Volterra integro-differential equations is presented under strongly generalised differentiability by applying an influent effective technique, called the Residual Power Series (RPS) method. The solution approach can be expressed on Taylor's series formula in terms of elementary σ-level representation, whereas the coefficients can be computed by utilising its residual functions. Furthermore, a numerical computational example is given to test and validate the proposed method. The results reached show several features concerning the RPS method such as potentiality, generality and superiority to handle many problems arising in physics and engineering.

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